and plugging this value of b into the first equation we get 4a + 14 = 5
whence

6. If a ball is thrown vertically upward with a velocity of 80 ft/s, then its
height after t seconds is h(t) = 80t - 16t2. What is the velocity of the
ball when it is 96 ft above the ground on its way up? On its way
down? (5 points)

Solution The velocity of the ball equals to the derivative of its height.

whence t = 2 or t = 3. At the moment t = 2 the velocity of the ball is
16 ft/sec and the ball is moving up whilst at the moment t = 3 the
velocity is -16 ft/sec and the ball is moving down.

7. An open box is to be made from a 6ft by 8ft rectangular piece of
sheet metal by cutting out squares of equal size from the four corners
and bending up the sides. Find the maximum value that the volume
of the box can have.(10 points)Solution Let x be the side of each cut out square. Then the height of
the box is x

whilst its length is 8-2x and its width is 6-2x (See the picture below).

Notice that because these dimensions cannot be negative we
have
0 ≤ x ≤ 3. There fore we have to maximize the volume of the box

V (x) = x(8-2x)(6-2x) = 4x(4-x)(3-x) = 4x(x2-7x+12) = 4(x3-7x2+12x)

on the interval [0, 3]. At the ends of the interval the function V takes
value 0 so it will take the greatest value at a stationary point inside
the interval. To find the stationary points we have to solve the
equation

The sign + provides a solution which is greater than 3 and
of no
interest to us. The only stationary point inside the interval [0, 3] is

You can easily check yourself that the greatest value of
the volume is

8. Sketch the graph of y = x3 - 3x + 2. Find x- and y-
intercepts, plot
the stationary points and the inflection points and determine the
intervals where y is increasing and decreasing, concave up and
concave down.(10 points)Solution. To find the x-intercepts we have to solve the equation

x3 - 3x + 2 = 0.

Notice that x = 1 is a solution of this equation and therefore x - 1 is
a factor of x3 - 3x + 2. We can factor x3 - 3x + 2 using long division,grouping , or synthetic division. The table below shows synthetic
division.

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